Exponent Calculator

What is an exponent?

An exponent tells you how many times to multiply a number, the base, by itself. In 2¹⁰ the base is 2 and the exponent (also called the power or index) is 10, so 2¹⁰ means ten 2s multiplied together, which is 1,024. Exponents appear everywhere: squares and cubes in area and volume, doubling and halving in computing (2¹⁰ bytes is a kibibyte), growth rates in compound interest, and scientific notation for very large and very small quantities.

Laws of exponents

The product and quotient rules only work when the bases are the same; 2³ × 3² cannot be simplified into a single power. They are also why the zero and negative rules must hold: 2³ ÷ 2³ is both 1 and 2⁰, and 2³ ÷ 2⁵ is both 1 ÷ 2² and 2⁻².

Worked examples

2¹⁰: double 1 ten times: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. In scientific notation that is 1.024 × 10³. The reciprocal, 1 ÷ 1,024 = 0.0009765625, is the same as 2⁻¹⁰.

Cube root of 27: ask which number multiplied by itself three times gives 27. Since 3 × 3 × 3 = 27, ³√27 = 3. Written with a fractional exponent, 27^(1/3) = 3, and the check 3³ = 27 confirms it.

Mixed cases: 5⁻² = 1 ÷ 25 = 0.04; (−2)³ = −2 × −2 × −2 = −8; 16^(3/4) = (⁴√16)³ = 2³ = 8; and 1.05¹⁰ ≈ 1.6289, which is how a 5% annual return grows over ten years.

Roots as fractional exponents

A root is just an exponent that is a fraction: ⁿ√x = x^(1/n), so the square root is the power ½ and the cube root is the power ⅓. More generally x^(p/q) means "take the q-th root, then raise to the p-th power". This is why the laws of exponents apply to roots too: √2 × √8 = √16 = 4 is the product rule in disguise. Odd roots of negative numbers are real (³√−27 = −3, because (−3)³ = −27), but even roots of negative numbers are not; √−4 has no real answer, which the calculator reports rather than inventing a value.

Scientific notation

Scientific notation writes a number as a mantissa between 1 and 10 multiplied by a power of ten: 1,024 = 1.024 × 10³ and 0.00052 = 5.2 × 10⁻⁴. The exponent counts how many places the decimal point moves, to the left for positive exponents and to the right for negative ones. Most calculators and programming languages show it as 1.024e3. It becomes essential as results grow: 2¹⁰⁰ is 1.2676506 × 10³⁰, a 31-digit number that would be unreadable written out.

How to use this calculator

1. Choose Power to raise a base to an exponent, or Root to take an nth root.
2. In power mode enter the base and the exponent. Negative, decimal and fractional exponents (enter 0.5 for ½) are all accepted.
3. In root mode enter the number and the root index: 2 for a square root, 3 for a cube root and so on.
4. Read the result, its scientific notation and either the reciprocal (for powers) or the check that raises the root back to the original number.

Common mistakes

• (−2)² versus −2²: the first squares −2 and gives 4; the second squares 2 and then negates, giving −4, because an exponent binds more tightly than a minus sign. Enter the base as −2 if you mean the first.
• A negative exponent does not make the result negative. 2⁻³ is 0.125, a positive fraction.
• x^(1/2) is the square root of x, not x ÷ 2. 16^(1/2) = 4, while 16 ÷ 2 = 8.
• Different bases cannot be combined with the product rule: 2³ × 5³ = (2 × 5)³ = 1,000 works only because the exponents match.
• 0⁰ and 0⁻ⁿ: this calculator returns 1 for 0⁰ (the common convention) and reports 0 to a negative power as undefined.
• Precision limits: results carry about 16 significant digits, so 2⁶⁴ = 18,446,744,073,709,551,616 is displayed as 1.844674407 × 10¹⁹. For exact large integers use the prime factorization calculator or a symbolic tool.